Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 320581, 10 pages
http://dx.doi.org/10.1155/2013/320581
Research Article
Analysis of a Dengue Disease Model with Nonlinear Incidence
Shu-Min Guo,1 Xue-Zhi Li,2 and Mini Ghosh3
1
Department of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China
Department of Mathematics, Xinyang Normal University, Xinyang 464000, China
3
School of Advanced Sciences, VIT University, Chennai Campus, Chennai 600048, India
2
Correspondence should be addressed to Xue-Zhi Li; xzli66@126.com
Received 10 September 2012; Revised 3 January 2013; Accepted 3 January 2013
Academic Editor: Eric Campos Canton
Copyright © 2013 Shu-Min Guo et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A dengue disease epidemic model with nonlinear incidence is formulated and analyzed. The equilibria and threshold of the model
are found. The stability of the system is analyzed through a geometric approach to stability. The proposed model also exhibits
backward bifurcation under suitable conditions on parameters. Our results imply that a nonlinear incidence produces rich dynamics
and they should be studied carefully in order to analyze the spread of disease more accurately. Finally, numerical simulations are
presented to illustrate the analytical findings.
1. Introduction
Dengue disease is transmitted in humans by mosquitoes,
called vectors, who carry the disease without getting it
themselves. Every year many people die from dengue disease.
It has become a global health problem due to its rapid spread.
In order to predict the development extent of such epidemics,
it is useful to find the suitable mathematical models, which
can provide insight into the dynamics of a disease and can
help us to make right decisions on public health policies.
However in the region of high infective prevalence such
as Africa, some tropical countries, and so forth, there are lots
of people infected with dengue, and so forth. and hence there
are lots of infected mosquitoes. So in these endemic regions
residents are frequently bitten by infected mosquitoes, thus
making individuals exposed with multiple bites of mosquito
which causes individuals to move to infected class with
increased rate compared to people who are exposed with
single effective bite of mosquito. So it is reasonable to consider
nonlinear saturation incidence in place of bilinear incidence
or standard incidence in the model as it also incorporates the
effects of increased exposure to mosquito bites.
In fact the incidence rate is very important in the transmission dynamics of the disease and the qualitative behaviour
of the disease changes with the change in the incidence rate.
In a recent paper by Xiao and Tang [1], interaction of the
nonlinear incidence and the partial immunity for a simple
SIV epidemic model is investigated and it is shown that due
to the nonlinear incidence, vaccination may contribute to
disease spread rather than to its elimination. A vector-host
epidemic model with nonlinear incidence is also analyzed
by Cai and Li [2]. Here they considered nonlinear saturating
type incidence and they studied the effect of this type of
incidence rate on the basic reproduction number and the
equilibrium level of the infective population. Hu et al. [3]
analyzed an SIR epidemic model with nonlinear incidence
rate and treatment. Here it is shown that reducing the basic
reproduction number below one is not enough to eliminate
the disease as backward bifurcation occurs for this model.
Also when the basic reproduction number is greater than
unity, there are multiple endemic equilibria. The conditions
for the stability of endemic equilibrium and the existence
of limit cycle are discussed in this paper. A discrete-time
epidemic model with nonlinear incidence rates is studied
by Li et al. [4], where a very complex dynamical behaviors
are demonstrated. It is shown that there are possibilities of
the transcritical bifurcation, flip bifurcation, Hopf bifurcation
and chaos. A vector-borne model with nonlinear incidence
is analyzed by Ozair et al. [5]. Here they considered the
conditions for the stability of endemic equilibrium.
2
Discrete Dynamics in Nature and Society
In recent years, dengue has become a major public
health problem and is endemic in several countries. It is
affecting almost one-third of the world’s population. Dengue
is an infection caused by a virus known as flavivirus and is
spread by their vector Aedes aegypti. There are four different
serotypes of flavivirus but infection with one serotype gives
life-long immunity to that serotype but not to other three
serotypes. So one can formulate dengue disease model as
SIS or SIR model depending upon serotypes of flavivirus.
In this paper we have formulated an SIR epidemic model
with nonlinear incidence assuming the infection of dengue
is caused by the flavivirus which is giving life-long immunity.
In dengue-endemic area a person can be infected with more
than one serotype of flavivirus in his/her lifetime. But in the
current study we have ignored this fact as it will complicate
the model and its analysis.
The paper is organized as follows. In Section 2 the model
and some preliminary properties are presented. In Section 3
equilibria and bifurcation are studied. Section 4 is devoted to
the global stability analysis of the endemic equilibrium, when
it is unique. Numerical simulations are given in Section 5, we
end the paper with conclusions in Section 6.
2. Model Formulation
The vector-host epidemic model with nonlinear incidence
can be described by the following system of differential
equations:
𝑑𝑆 (𝑡)
= 𝑏1 − 𝜆 1 𝑆 (𝑡) 𝑉 (𝑡) [1 + 𝛼𝑉 (𝑡)] − 𝜇1 𝑆 (𝑡) ,
𝑑𝑡
𝑑𝑁1 (𝑡)
= 𝑏1 − 𝜇1 𝑁1 (𝑡) ,
𝑑𝑡
(2)
which is derived by adding the first three equations in (1). The
total number of vectors 𝑁2 (𝑡) can be determined by 𝑁2 (𝑡) =
𝑀(𝑡) + 𝑉(𝑡) or from the differential equation
𝑑𝐼 (𝑡)
= 𝜆 1 𝑆 (𝑡) 𝑉 (𝑡) [1 + 𝛼𝑉 (𝑡)] − 𝛾𝐼 (𝑡) − 𝜇1 𝐼 (𝑡) ,
𝑑𝑡
𝑑𝑅 (𝑡)
= 𝛾𝐼 (𝑡) − 𝜇1 𝑅 (𝑡) ,
𝑑𝑡
new infective individuals arise from double exposures at a
rate 𝛼𝜆 1 𝑆(𝑡)𝑉2 (𝑡).
The second component of the vector population is the
number of pathogen-free (susceptible) vectors at time 𝑡, given
by 𝑀(𝑡). The total size of the vector population at time 𝑡, given
by 𝑁2 (𝑡) and is subdivided into these two vector-population
classes, the susceptible vectors and infectious vectors. The
vector population dies at a natural death rate 𝜇2 . In addition,
the vector population is recruited at a birth rate 𝑏2 . Although
there are evidences that the pathogen of several vectorborne diseases (e.g., West Nile fever, yellow fever and Lyme
disease) can be transmitted from (female) parent to offspring
in the vector population, we will assume that all newborn
vectors are susceptible and vertical transmission is neglected.
Susceptible vectors start carrying the pathogen after getting
into contact (biting) an infective host at a rate 𝜆 2 so that the
incidence of newly infected vectors is given by a mass-action
term 𝜆 2 𝑀(𝑡)𝐼(𝑡). In contrast to the host population, once the
vectors become carriers of the microparasite, they carry it for
life.
We make some reasonable technical assumptions on the
parameters of the model, namely that 𝛾 > 0, 𝜇𝑗 > 0 and 𝑏𝑗 > 0
for 𝑗 = 1, 2. The above systems for the host population and
the vector are also equipped with initial conditions as follows:
𝑆(0) = 𝑆0 , 𝐼(0) = 𝐼0 , 𝑅(0) = 𝑅0 , 𝑀(0) = 𝑀0 and 𝑉(0) = 𝑉0 .
The total population size 𝑁1 (𝑡) can be determined by
𝑁1 (𝑡) = 𝑆(𝑡) + 𝐼(𝑡) + 𝑅(𝑡) or from the differential equation
𝑑𝑁2 (𝑡)
= 𝑏2 − 𝜇2 𝑁2 ,
𝑑𝑡
(1)
𝑑𝑀 (𝑡)
= 𝑏2 − 𝜆 2 𝑀 (𝑡) 𝐼 (𝑡) − 𝜇2 𝑀 (𝑡) ,
𝑑𝑡
𝑑𝑉 (𝑡)
= 𝜆 2 𝑀 (𝑡) 𝐼 (𝑡) − 𝜇2 𝑉 (𝑡) .
𝑑𝑡
The total host population size at time 𝑡, given by 𝑁1 (𝑡), is
partitioned into subclasses of individuals who are susceptible,
infectious and recovered, with sizes denoted by 𝑆(𝑡), 𝐼(𝑡) and
𝑅(𝑡), respectively. Furthermore, the host population dies at
a natural death rate 𝜇1 . In addition, the host population is
recruited at a rate 𝑏1 . We assume that vertical transmission in
the host does not occur so that all newly recruited individuals
are susceptible. The per capita recovery rate of the hosts
is given by 𝛾. The recovered individuals are assumed to
acquire permanent immunity and there is no transfer from
the 𝑅 class back to the 𝑆 class. The nonlinear incidence
rate is 𝜆 1 𝑆(𝑡)𝑉(𝑡)[1 + 𝛼𝑉(𝑡)], where 𝜆 1 and 𝛼 are positive
constants, 𝑉(𝑡) is the number of vectors at time 𝑡 who carry
the pathogen. It corresponds to an increased rate of infection
due to two exposures over a short time period. The single
contacts lead to infection at the rate 𝜆 1 𝑆(𝑡)𝑉(𝑡), whereas the
(3)
which is derived by adding the last two equations in (1).
It is easily seen that both for the host population and
for the vector population the corresponding total population
sizes are asymptotically constant:
lim 𝑁1 (𝑡) =
𝑡→∞
𝑏1
,
𝜇1
lim 𝑁2 (𝑡) =
𝑡→∞
𝑏2
.
𝜇2
(4)
So we can assume without loss of generality that 𝑁1 (𝑡) =
𝑏1 /𝜇1 , 𝑁2 (𝑡) = 𝑏2 /𝜇2 for all 𝑡 ≥ 0.
Then the dynamics of system (1) is qualitatively equivalent
to the dynamics of system given by
𝑑𝑆 (𝑡)
= 𝑏1 − 𝜆 1 𝑆 (𝑡) 𝑉 (𝑡) [1 + 𝛼𝑉 (𝑡)] − 𝜇1 𝑆 (𝑡) ,
𝑑𝑡
𝑑𝐼 (𝑡)
= 𝜆 1 𝑆 (𝑡) 𝑉 (𝑡) [1 + 𝛼𝑉 (𝑡)] − 𝛾𝐼 (𝑡) − 𝜇1 𝐼 (𝑡) ,
𝑑𝑡
(5)
𝑏
𝑑𝑉 (𝑡)
= 𝜆 2 [ 2 − 𝑉 (𝑡)] 𝐼 (𝑡) − 𝜇2 𝑉 (𝑡) .
𝑑𝑡
𝜇2
The values of 𝑅 and 𝑀 can be determined correspondingly
from 𝑅 = 𝑏1 /𝜇1 − 𝑆 − 𝐼, and 𝑀 = 𝑏2 /𝜇2 − 𝑉 respectively.
Discrete Dynamics in Nature and Society
3
For biological reasons we need the solutions nonnegative.
Mathematical properties of the solutions lead us to study the
system (5) in the closed set
Γ = {(𝑆, 𝐼, 𝑉) ∈ R+3 | 0 ≤ 𝑆 + 𝐼 ≤
𝑏1
𝑏
, 0 ≤ 𝑉 ≤ 2 , 𝑆 ≥ 0, 𝐼 ≥ 0} ,
𝜇1
𝜇2
(6)
where R+3 denotes the non-negative cone of R3 including its
lower dimensional faces. It can be verified that Γ is positively
invariant with respect to (5). We denote by 𝜕Γ and Γ0 the
boundary and the interior of Γ in R3 , respectively.
Furthermore, we observe that there is a bifurcation point
when 𝑎3 > 0, 𝑎2 < 0 and 𝑎22 − 4𝑎1 𝑎3 = 0. In fact the quantity
𝑎22 − 4𝑎1 𝑎3 can be expressed in terms of new threshold 𝑅0∗ ,
where
𝑅0∗ =
𝑀0 =
𝑀0
,
𝑁0
2𝜆21 𝜆22 𝑏12 𝑏2 𝛼
2𝜆2 𝜆 𝑏 𝑏 𝛼
+ 1 2 1 2 + 𝑃0 ,
𝜇1
𝜇1 𝜇2 (𝛾 + 𝜇1 )
𝜆 1 𝜆 2 𝑏1 𝜇22 + 2𝜆 1 𝜇23 (𝛾 + 𝜇1 ) 𝜆 1 𝜇24 (𝛾 + 𝜇1 )
+
𝑏2
𝜆 2 𝑏1 𝑏2
3. Equilibria and Bifurcation Analysis
+ 𝜆 1 𝜆 2 𝑏1 𝑏2 𝛼2 + 𝑃0 ,
The disease-free equilibrium of system (5) is 𝐸0 = (𝑏1 /
𝜇1 , 0, 0). The endemic equilibria 𝐸1 = (𝑆∗ , 𝐼∗ , 𝑉∗ ) of system
(5) can be deduced by the system,
𝑃0 = 4𝜆 1 𝜆 2 𝑏1 𝛼𝜇2 + 4𝜆 1 𝛼𝜇22 (𝛾 + 𝜇1 ) .
𝑏1 − 𝜆 1 𝑆∗ 𝑉∗ [1 + 𝛼𝑉∗ ] − 𝜇1 𝑆∗ = 0,
𝜆 1 𝑆∗ 𝑉∗ [1 + 𝛼𝑉∗ ] − 𝛾𝐼∗ − 𝜇1 𝐼∗ = 0,
(7)
𝑏
𝜆 2 [ 2 − 𝑉∗ ] 𝐼∗ − 𝜇2 𝑉∗ = 0,
𝜇2
which gives,
𝑆∗ =
𝑏1
,
𝜆 1 𝑉∗ (1 + 𝛼𝑉∗ ) + 𝜇1
𝐼∗ =
𝜇2 𝑉∗
,
𝜆 2 (𝑏2 /𝜇2 − 𝑉∗ )
𝑎1 𝑉
∗
+ 𝑎2 𝑉 + 𝑎3 = 0,
By a little algebraic calculation, we have 𝑎22 − 4𝑎1 𝑎3 > 0 whenever 𝑅0 > 𝑅0∗ . Taking into account the above consideration,
we have the following theorem.
Theorem 1. System (7) admits a unique endemic equilibrium
𝐸 when 𝑅0 ≥ 1; there are no endemic equilibria for 𝑅0 < 𝑅0∗ ;
there are two endemic equilibria, 𝐸1 and 𝐸2 , for 𝑅 < 𝑅0 < 1
and 𝑅 = max{𝑅0∗ , 𝑅1∗ }.
Next we analyze the stability of the disease-free equilibrium 𝐸0 = (𝑏1 /𝜇1 , 0, 0). The Jacobian matrix corresponding
to (7) is
(8)
and 𝑉∗ is the positive solution of the following equation,
∗2
𝐽 (𝑆, 𝐼, 𝑉)
−𝜆 1 𝑉 (1 + 𝛼𝑉)−𝜇1
0
−𝜆 1 𝑆 (1+2𝛼𝑉)
𝜆 1 𝑉 (1+𝛼𝑉)
−𝛾−𝜇1
𝜆 1 𝑆 (1+2𝛼𝑉)
(9)
=(
where
0
𝑎1 = 𝜆 1 𝜆 2 𝑏1 𝛼 + 𝜇2 𝜆 1 𝛼 (𝛾 + 𝜇1 ) ,
𝑎2 = 𝜆 1 𝜆 2 𝑏1 + 𝜇2 𝜆 1 (𝛾 + 𝜇1 ) −
𝜆2 (
𝜆 1 𝜆 2 𝑏1 𝑏2 𝛼
𝜇2
𝑏2
−𝑉)
𝜇2
𝑅1∗ =
(13)
𝜆 1 𝜆 2 𝑏1 𝑏2
= 𝜇1 𝜇2 (𝛾 + 𝜇1 ) (1 − 𝑅0 ) ,
𝜇2
𝜆 1 𝜆 2 𝑏1 + 𝜇2 𝜆 1 (𝛾 + 𝜇1 )
,
𝛼𝜇1 𝜇2 (𝛾 + 𝜇1 )
𝑅0 =
𝜆 1 𝜆 2 𝑏1 𝑏2
.
𝜇1 𝜇22 (𝛾 + 𝜇1 )
𝑎2 < 0 ⇐⇒ 𝑅1∗ < 𝑅0 ;
𝑎3 > 0 ⇐⇒ 𝑅0 < 1.
First, we observe that
−𝜇1
𝐽(
(10)
𝑏1
(
, 0, 0) = ( 0
𝜇1
(
0
0
−𝛾 − 𝜇1
𝜆2
𝑏2
𝜇2
𝑏1
𝜇1
𝑏1 )
𝜆1
),
𝜇1
−𝜆 1
−𝜇2
(14)
)
so that the eigenvalues 𝜉 are given by the roots of the following
qubic equation
Thus, we observe that
𝑎1 > 0;
).
−𝜆 2 𝐼−𝜇2
= 𝛼𝜇1 𝜇2 (𝛾 + 𝜇1 ) (𝑅1∗ − 𝑅0 ) ,
𝑎3 = 𝜇1 𝜇2 (𝛾 + 𝜇1 ) −
(12)
2
𝑁0 =
(11)
By the Descartes’ rule of signs, it follows that there is a unique
endemic equilibrium whenever 𝑎3 < 0 and there are two endemic equilibria whenever 𝑎3 > 0, 𝑎2 < 0 and 𝑎22 − 4𝑎1 𝑎3 > 0.
(𝜇1 + 𝜉) [𝜉2 + (𝛾 + 𝜇1 + 𝜇2 ) 𝜉 + 𝜇2 (𝛾 + 𝜇1 ) (1 − 𝑅0 )] = 0.
(15)
Hence if 𝑅0 < 1, then the eigenvalues are all negatives
and 𝐸0 is locally asymptotically stable. If 𝑅0 > 1, then two
4
Discrete Dynamics in Nature and Society
eigenvalues are negative and one is positive, so that 𝐸0 is
unstable.
Since there are no explicit expressions for equilibria 𝐸1
and 𝐸2 and the existence conditions for the two equilibria
are very complex, it is very difficult to prove their stability by
using the existing analytic methods.
Theorem 1 establishes that 𝑅0 = 1 is a bifurcation value.
In fact, across 𝑅0 = 1 the disease free equilibrium changes
its stability properties. Now, we investigate what kind of
bifurcation occurs at 𝑅0 = 1. In order to do that, we will make
use of the result summarized below, which has been obtained
in [6] and is based on the use of the center manifold theory
[7].
Let us consider a general system of ODEs with a parameter 𝜙:
𝑛
𝑥̇ = 𝑓 (𝑥, 𝜙) ;
𝑛
𝑓 : 𝑅 × 𝑅 󳨀→ 𝑅 ,
2
It clearly appears that, at 𝜙 = 0 a transcritical bifurcation
takes place: more precisely, when 𝑎 < 0 and 𝑏 > 0, such a
bifurcation is forward; when 𝑎 > 0 and 𝑏 > 0 the bifurcation
at 𝜙 = 0 is backward. We will apply Theorem 2 to show that
system (7) may exhibit a backward bifurcation when 𝑅0 = 1.
We consider the disease-free equilibrium 𝐸0 = (𝑏1 /𝜇1 , 0, 0)
and observe that the condition 𝑅0 = 1 can be seen, in terms
of the parameter 𝜆 1 , as 𝜆 1 = 𝜆∗ = 𝜇1 𝜇22 (𝛾 + 𝜇1 )/𝜆 2 𝑏1 𝑏2 . The
eigenvalues of the matrix
−𝜇1
(
𝐽 (𝐸0 , 𝜆∗ ) = ( 0
(
𝑛
𝑓 ∈ 𝐶 (𝑅 × 𝑅) .
(16)
Without loss of generality, we assume that 𝑥 = 0 is an
equilibrium for (16).
Theorem 2 (see (Castillo-Chavez and Song [6])). Assume:
(A1) 𝐴 = 𝐷𝑥 𝑓(0, 0) is the linearization matrix of system
(16) around the equilibrium 𝑥 = 0 with 𝜙 evaluated
at 0. Zero is a simple eigenvalue of 𝐴 and all other
eigenvalues of 𝐴 have negative real parts.
𝜉1 = −𝜇1 ,
𝑎 = ∑ 𝜃𝑘 𝜔𝑖 𝜔𝑗
𝑘,𝑖,𝑗
𝜕 𝑓𝑘
(0, 0) ,
𝜕𝑥𝑖 𝜕𝑥𝑗
𝑛
𝑏 = ∑ 𝜃𝑘 𝜔𝑖
𝑘,𝑖,𝑗
(iii) 𝑎 > 0, 𝑏 < 0. When 𝜙 < 0, with |𝜙| ≪ 1, 𝑥 = 0 is
unstable and there exists a locally asymptotically stable
negative equilibrium; when 0 < |𝜙| ≪ 1, 𝑥 = 0 is stable
and a positive unstable equilibrium appears;
(iv) 𝑎 < 0, 𝑏 > 0. When 𝜙 changes from negative to positive,
𝑥 = 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes
positive and locally asymptotically stable.
)
𝜉3 = 0.
(19)
𝜇22 (𝛾 + 𝜇1 )
𝜔3 = 0,
𝜆 2 𝑏2
(20)
𝜆 2 𝑏2
𝜔 − 𝜇2 𝜔3 = 0,
𝜇2 2
(17)
(ii) 𝑎 < 0, 𝑏 < 0. When 𝜙 < 0, with |𝜙| ≪ 1, 𝑥 =
0 is unstable; when 0 < |𝜙| ≪ 1, 𝑥 = 0 is
locally asymptotically stable and there exists a positive
unstable equilibrium;
−𝜇2
𝜇22 (𝛾 + 𝜇1 )
𝜔3 = 0,
𝜆 2 𝑏2
− (𝛾 + 𝜇1 ) 𝜔2 +
𝜕 𝑓𝑘
(0, 0) .
𝜕𝑥𝑖 𝜕𝜙
(i) 𝑎 > 0, 𝑏 > 0. When 𝜙 < 0, with |𝜙| ≪ 1, 𝑥 =
0 is locally asymptotically stable and there exists a
positive unstable equilibrium; when 0 < |𝜙| ≪ 1,
𝑥 = 0 is unstable and there exists a negative and locally
asymptotically stable equilibrium;
𝑏2
𝜇2
𝜉2 = −𝛾 − 𝜇1 − 𝜇2 ,
−𝜇1 𝜔1 −
2
Then the local dynamics of system (16) around 𝑥 = 0 are totally
determined by 𝑎 and 𝑏
𝜆2
(18)
Thus 𝜉3 = 0 is a simple zero eigenvalue of the matrix
𝐽(𝐸0 , 𝜆∗ ) and the other eigenvalues are real and negative.
Hence, when 𝜆 1 = 𝜆∗ (or equivalently when 𝑅0 = 1), the
disease-free equilibrium 𝐸0 is a nonhyperbolic equilibrium:
the assumption (A1) of Theorem 2 is then verified.
Now we denote by 𝜔 = (𝜔1 , 𝜔2 , 𝜔3 )𝑇 , a right eigenvector
associated with the zero eigenvalue 𝜉3 = 0. It follows:
Let 𝑓𝑘 denotes the 𝑘th component of 𝑓 and,
2
−𝛾 − 𝜇1
0
𝑏1
𝜇1
𝑏
𝜆∗ 1 )
),
𝜇1
−𝜆∗
are given by
(A2) Matrix 𝐴 has a right eigenvector 𝜔 and a left eigenvector 𝜃 corresponding to the zero eigenvalue.
𝑛
0
so that
𝑇
𝜔 = (−
𝛾 + 𝜇1
𝜆 𝑏
, 1, 22 2 ) .
𝜇1
𝜇2
(21)
Furthermore, the left eigenvector 𝜃 = (𝜃1 , 𝜃2 , 𝜃3 ) satisfying
𝜔 ⋅ 𝜃 = 1 is given by:
−𝜇1 𝜃1 = 0,
− (𝛾 + 𝜇1 ) 𝜃2 +
𝜆 2 𝑏2
𝜃 = 0,
𝜇2 3
𝜇22 (𝛾 + 𝜇1 )
(𝜃2 − 𝜃1 ) − 𝜇2 𝜃3 = 0,
𝜆 2 𝑏2
𝜃3 (
(22)
𝜆 2 𝑏2
𝜆 𝑏
+ 22 2 ) = 1.
𝜇2
𝜇2 (𝛾 + 𝜇1 )
Then, the left eigenvector 𝜃 turns out to be:
𝜃 = (0,
𝜇22 (𝛾 + 𝜇1 )
𝜇2
,
).
𝛾 + 𝜇1 + 𝜇2 𝜆 2 𝑏2 (𝛾 + 𝜇1 + 𝜇2 )
(23)
Discrete Dynamics in Nature and Society
5
We can thus compute the coefficient 𝑎 and 𝑏 defined in
Theorem 2, that is,
3
𝜕2 𝑓𝑘
𝑎 = ∑ 𝜃𝑘 𝜔𝑖 𝜔𝑗
(𝐸0 , 𝜆∗ ) ,
𝜕𝑥
𝜕𝑥
𝑖
𝑗
𝑘,𝑖,𝑗=1
3
𝜕2 𝑓𝑘
𝑏 = ∑ 𝜃𝑘 𝜔𝑖
(𝐸0 , 𝜆∗ ) .
𝜕𝑥
𝜕𝜆
𝑖
1
𝑘,𝑖=1
(24)
Taking into account of system (5) and considering in 𝑎
and 𝑏 only the nonzero derivatives for the terms (𝜕2 𝑓𝑘 /
𝜕𝑥𝑖 𝜕𝑥𝑗 )(𝐸0 , 𝜆∗ ) and (𝜕2 𝑓𝑘 /𝜕𝑥𝑖 𝜕𝜆 1 )(𝐸0 , 𝜆∗ ), it follows that:
𝑎 = 2𝜃1 𝜔1 𝜔3
+ 2𝜃3 𝜔2 𝜔3
lim inf 𝑆 (𝑡) > 𝑐,
𝑡 → +∞
𝜕2 𝑓3
(𝐸 , 𝜆∗ ) ,
𝜕𝑉𝜕𝐼 0
lim inf 𝐼 (𝑡) > 𝑐,
𝑡 → +∞
(28)
lim inf 𝑉 (𝑡) > 𝑐.
𝑡 → +∞
(25)
𝜕2 𝑓1
𝜕2 𝑓1
(𝐸0 , 𝜆∗ ) + 𝜃1 𝜔3
(𝐸 , 𝜆∗ )
𝜕𝑆𝜕𝜆 1
𝜕𝑉𝜕𝜆 1 0
+ 𝜃2 𝜔1
In Section 3 it has been shown that 𝑅0 ≥ 1 implies the
existence and uniqueness of the endemic equilibrium 𝐸. The
stability analysis of 𝐸 will be here performed through the geometric approach to global stability due to Li and Muldowney
[8]. The method has been summarized in the Appendix.
System (5), under the assumption 𝑅0 > 1, satisfies
conditions (H1) and (H2) in Appendix. In fact, when 𝑅0 >
1, then 𝐸0 is unstable. The instability of 𝐸0 , together with
𝐸0 ∈ 𝜕Γ, implies the uniform persistence, that is, there exists
a constant 𝑐 > 0 such that:
𝜕2 𝑓1
𝜕2 𝑓
(𝐸0 , 𝜆∗ ) + 𝜃1 𝜔32 21 (𝐸0 , 𝜆∗ )
𝜕𝑆𝜕𝑉
𝜕𝑉
𝜕2 𝑓2
𝜕2 𝑓
+ 2𝜃2 𝜔1 𝜔3
(𝐸0 , 𝜆∗ ) + 𝜃2 𝜔32 22 (𝐸0 , 𝜆∗ )
𝜕𝑆𝜕𝑉
𝜕𝑉
𝑏 = 𝜃1 𝜔1
4. Global Stability Analysis
𝜕2 𝑓2
𝜕2 𝑓2
(𝐸0 , 𝜆∗ ) + 𝜃2 𝜔3
(𝐸 , 𝜆∗ ) .
𝜕𝑆𝜕𝜆 1
𝜕𝑉𝜕𝜆 1 0
The uniform persistence is equivalent to the existence of a
compact set in the interior of Γ which is absorbing for (5).
Thus, (H1) is verified. Moreover, as previously shown, 𝐸 is
the only equilibrium in the interior of Γ, so that (H2) is
verified, too. Now it remains to find conditions for which the
Bendixson criterion given by Theorem A.4 is verified. Taking
into account of the Jacobian matrix (13), we obtain the second
additive compound matrix 𝐽[2] (𝑆, 𝐼, 𝑉),
[2]
𝐽11
In view of (21) and (23), we get
𝐽[2]
2𝜆 2
𝑎=
[𝜆 𝜆 𝑏 𝑏2 𝛼
3
𝜇1 𝜇2 (𝛾 + 𝜇1 + 𝜇2 ) 1 2 1 2
𝑏2
= (𝜆 2 ( − 𝑉)
𝜇2
0
𝜆 1 𝑆 (1 + 2𝛼𝑉) 𝜆 1 𝑆 (1 + 2𝛼𝑉)
[2]
𝐽22
0
𝜆 1 𝑉 (1 + 𝛼𝑉)
[2]
𝐽33
(29)
− 𝜇22 (𝛾 + 𝜇1 ) (𝜆 1 𝑏2 + 𝜇1 𝜇2 )] ,
𝑏=
𝜆 2 𝑏1 𝑏2
.
𝜇1 𝜇2 (𝛾 + 𝜇1 + 𝜇2 )
where
𝛿=
𝜇22 (𝛾 + 𝜇1 ) (𝜆 1 𝑏2 + 𝜇1 𝜇2 )
[2]
𝐽11
= −𝜆 1 𝑉 (1 + 𝛼𝑉) − 𝛾 − 2𝜇1 ,
(26)
Observe that the coefficient 𝑏 is always positive so that, according to Theorem 2, it is the sign of the coefficient 𝑎
which decides the local dynamics around the disease-free
equilibrium for 𝜆 1 = 𝜆∗ .
Let us introduce
𝜆 1 𝜆 2 𝑏1 𝑏22 𝛼
),
.
[2]
= −𝜆 1 𝑉 (1 + 𝛼𝑉) − 𝜇1 − 𝜆 2 𝐼 − 𝜇2 ,
𝐽22
[2]
= −𝛾 − 𝜇1 − 𝜆 2 𝐼 − 𝜇2 .
𝐽33
We consider the following function
𝐼 𝐼 𝐼
𝑃 = 𝑃 (𝑆, 𝐼, 𝑉) = diag { , , } .
𝑉 𝑉 𝑉
(27)
The coefficient 𝑎 is positive if and only if 𝛿 > 1. In this case,
the direction of the bifurcation of system (5) at 𝑅0 = 1 is
backward. So we summarize the above results in the following
theorem.
Theorem 3. If 𝛿 > 1, system (5) exhibits a backward bifurcation when 𝑅0 = 1. If 𝛿 < 1, system (5) exhibits a forward
bifurcation when 𝑅0 = 1.
The local dynamics in the neighborhood of the bifurcation value 𝑅0 = 1 is described in the former case by the (i)
of Theorem 2 and in the later case by the (iv) of the same
theorem.
(30)
(31)
So that
𝑃𝑓 = diag {
𝑃𝑓 𝑃
−1
𝐼 󸀠 𝑉 − 𝑉󸀠 𝐼 𝐼 󸀠 𝑉 − 𝑉󸀠 𝐼 𝐼 󸀠 𝑉 − 𝑉󸀠 𝐼
,
,
},
𝑉2
𝑉2
𝑉2
𝐼 󸀠 𝑉󸀠 𝐼󸀠 𝑉󸀠 𝐼󸀠 𝑉󸀠
= diag { − , − , − } .
𝐼
𝑉 𝐼
𝑉 𝐼
𝑉
(32)
Therefore
𝐵 𝐵
𝐵 = 𝑃𝑓 𝑃−1 + 𝑃𝐽[2] 𝑃−1 = ( 11 12 ) ,
𝐵21 𝐵22
(33)
6
Discrete Dynamics in Nature and Society
where
𝐵11 =
Substituting (40) into (38) and (40) into (39), respectively, we
have
𝐼󸀠 𝜆 𝑏 𝐼
𝑔1 = − 2 2 + 𝜆 2 𝐼 + 𝜇2 − 𝜆 1 𝑉 (1 + 𝛼𝑉)
𝐼
𝜇2 𝑉
𝐼󸀠 𝑉󸀠
−
− 𝜆 1 𝑉 (1 + 𝛼𝑉) − 2𝜇1 − 𝛾,
𝐼
𝑉
𝐵12 = [𝜆 1 𝑆 (1 + 2𝛼𝑉) , 𝜆 1 𝑆 (1 + 2𝛼𝑉)] ,
𝐵21 = [𝜆 2 (
− 2𝜇1 − 𝛾 + 𝜆 1 𝑆 (1 + 2𝛼𝑉)
𝑇
𝑏2
− 𝑉) , 0] ,
𝜇2
≤
𝐵22
𝐼󸀠 𝑉󸀠
0
[ − −𝜆 1 𝑉 (1+𝛼𝑉)−𝜇1 −𝜆 2 𝐼−𝜇2
]
𝐼 𝑉
].
=[
[
]
𝐼󸀠 𝑉󸀠
𝜆 1 𝑉 (1 + 𝛼𝑉)
− −𝛾−𝜇1 −𝜆 2 𝐼−𝜇2
[
]
𝐼 𝑉
𝑔2 =
≤
(34)
(35)
󵄨 󵄨
󵄨 󵄨
𝜇 (𝐵) ≤ sup {𝑔1 , 𝑔2 } = sup {𝜇1 (𝐵11 )+ 󵄨󵄨󵄨𝐵12 󵄨󵄨󵄨 , 𝜇1 (𝐵22 )+ 󵄨󵄨󵄨𝐵21 󵄨󵄨󵄨} ,
(36)
where |𝐵12 |, |𝐵21 | are matrix norms with respect to the 𝑙1
vector norm, and 𝜇1 denotes the Lozinskii ̆ measure with
respect to the 𝑙1 norm. More specifically,
𝑏
+ 𝜆 1 𝑆 (1 + 2𝛼𝑉) , 𝜆 2 ( 2 − 𝑉)} .
𝜇2
𝑏1
𝑏
𝑏
+ 𝜇2 + 𝜆 1 1 (1 + 2𝛼 2 ) < 𝜆 1 𝑐 (1 + 𝛼𝑐) + 𝜇1 + 𝛾,
𝜇1
𝜇1
𝜇2
(44)
𝜇 (𝐵) ≤
𝐼󸀠
+ Π,
𝐼
(45)
where
Π = max {𝜆 2
𝑏1
𝑏
𝑏
+ 𝜇2 + 𝜆 1 1 (1 + 2𝛼 2 ) − 𝜆 1 𝑐 (1 + 𝛼𝑐)
𝜇1
𝜇1
𝜇2
− 𝜇1 − 𝛾, 𝜆 2 (
𝐼󸀠 𝑉󸀠
𝑔1 = −
− 𝜆 1 𝑉 (1 + 𝛼𝑉) − 2𝜇1 − 𝛾 + 𝜆 1 𝑆 (1 + 2𝛼𝑉) ,
𝐼
𝑉
(38)
(39)
Observe that system (5) provides the following equalities
𝑉󸀠 𝜆 2 𝑏2 𝐼
=
− 𝜆 2 𝐼 − 𝜇2 .
𝑉
𝜇2 𝑉
(43)
In view of 𝑐 ≤ 𝑆, 𝐼 ≤ 𝑏1 /𝜇1 and 𝑐 ≤ 𝑉 ≤ 𝑏2 /𝜇2 , we can deduce
that if
𝑏
𝜆 2 ( 2 − 𝑐) < 𝜇1 ,
𝜇2
(37)
𝐼󸀠 𝑉󸀠
−
− 𝜇1 − 𝜆 2 𝐼 − 𝜇2 .
𝐼
𝑉
𝑏
𝐼󸀠 𝑉󸀠
−
− 𝜇1 − 𝜆 2 𝐼 − 𝜇2 + 𝜆 2 ( 2 − 𝑉) .
𝐼
𝑉
𝜇2
(42)
Λ = max {𝜆 2 𝐼 + 𝜇2 − 𝜆 1 𝑉 (1 + 𝛼𝑉) − 𝜇1 − 𝛾
Therefore,
𝑔2 =
𝐼󸀠
− 𝜇1 + Λ,
𝐼
then
2
𝜇1 (𝐵22 ) =
(41)
where
𝜆2
𝐼󸀠 𝑉󸀠
−
− 𝜆 1 𝑉 (1 + 𝛼𝑉) − 𝛾 − 2𝜇1 ,
𝐼
𝑉
󵄨󵄨 󵄨󵄨
󵄨󵄨𝐵12 󵄨󵄨 = 𝜆 1 𝑆 (1 + 2𝛼𝑉) ,
𝑏
󵄨󵄨 󵄨󵄨
󵄨󵄨𝐵21 󵄨󵄨 = 𝜆 2 ( 2 − 𝑉) ,
𝜇
𝑏
𝐼󸀠
− 𝜇1 + 𝜆 2 ( 2 − 𝑉) .
𝐼
𝜇2
𝜇 (𝐵) ≤
where (𝑢1 , 𝑢2 , 𝑢3 ) denotes the vector in 𝑅3 and denote by 𝜇
the Lozinskii ̆ measure with respect to this norm. It follows
𝜇1 (𝐵11 ) =
𝑏
𝐼󸀠 𝜆 2 𝑏2 𝐼
−
− 𝜇1 + 𝜆 2 ( 2 − 𝑉)
𝐼
𝜇2 𝑉
𝜇2
Thus, (36) implies
Consider now the norm in 𝑅3 as
󵄨 󵄨 󵄨 󵄨 󵄨 󵄨
󵄨
󵄨󵄨
󵄨󵄨𝑢1 , 𝑢2 , 𝑢3 󵄨󵄨󵄨 = max {󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑢3 󵄨󵄨󵄨} ,
𝐼󸀠
+ 𝜆 2 𝐼 + 𝜇2 − 𝜆 1 𝑉 (1+𝛼𝑉) − 2𝜇1 − 𝛾 + 𝜆 1 𝑆 (1+2𝛼𝑉) ,
𝐼
(40)
𝑏2
− 𝑐) − 𝜇1 } < 0.
𝜇2
(46)
Hence
1
𝐼 (𝑡)
1 𝑡
+ Π,
∫ 𝜇 (𝐵) 𝑑𝑠 ≤ log
𝑡 0
𝑡
𝐼 (0)
(47)
and the Bendixson criterion given by Theorem A.4 is thus
verified. The discussions above may be summarized as follows.
Theorem 4. If 𝑅0 ≥ 1, then system (5) admits a unique
endemic equilibrium 𝐸. It is globally asymptotically stable with
respect to solutions of (5) initiating in the interior of Γ, provided
that inequality (44) holds true.
Discrete Dynamics in Nature and Society
7
3000
2000
2500
1500
2000
𝐼
𝐼
1500
1000
1000
500
500
0
0
500
1000
𝑆
1500
0
2000
Figure 1: Phase plot of 𝐼 verses 𝑆 showing bistability when 𝑅 < 𝑅0 <
1 and 𝛿 > 1 for the parameter values 𝑏1 = 240, 𝜆 1 = 0.00012, 𝛼 =
0.08, 𝜇1 = 0.16, 𝛾 = 0.015, 𝜆 2 = 0.0001, 𝑏2 = 550, 𝜇2 = 0.5.
0
500
1000
1500
𝑆
2000
2500
3000
Figure 3: Phase plot of 𝐼 verses 𝑆 showing stability of unique feasible
disease free equilibrium point when 𝑅0 < 𝑅0∗ < 1 for the parameter
values 𝑏1 = 80, 𝜆 1 = 0.00001, 𝛼 = 0.04, 𝜇1 = 0.06, 𝛾 = 0.015, 𝜆 2 =
0.0002, 𝑏2 = 200, 𝜇2 = 0.5.
3000
1000
2500
800
2000
𝐼
𝐼
1500
400
1000
500
0
600
200
0
500
1000
1500
2000
𝑆
Figure 2: Phase plot of 𝐼 verses 𝑆 showing stability of unique
endemic equilibrium point when 𝑅0 > 1 and 𝛿 > 1 for the parameter
values 𝑏1 = 80, 𝜆 1 = 0.0004, 𝛼 = 0.04, 𝜇1 = 0.06, 𝛾 = 0.015, 𝜆 2 =
0.0002, 𝑏2 = 200, 𝜇2 = 0.5.
0
0
500
1000
1500
2000
𝑆
Figure 4: Phase plot of 𝐼 verses 𝑆 showing stability of unique feasible
endemic equilibrium point and instablity of disease free equilibrium
point when 𝛿 < 1, 𝑅0 > 1 for the parameter values 𝑏1 = 10, 𝜆 1 =
0.0005, 𝛼 = 0.01, 𝜇1 = 0.01, 𝛾 = 0.01, 𝜆 2 = 0.0003, 𝑏2 = 60, 𝜇2 =
0.5.
5. Simulation
The system (5) is simulated for various sets of parameters
using the package XPP [9]. In Figures 1–4, (𝑆, 𝐼) phase planes
are drawn which confer the existence and the stability of
different equilibria of the system (5). Here Figure 1 corresponds to the situation stated in Theorem 1, where 𝑅 < 𝑅0 <
1 and we get two endemic equilibria 𝐸1 and 𝐸2 and one
disease free equilibria 𝐸0 (1500, 0, 0). And for 𝛿 > 1, it is
found that disease free equilibrium 𝐸0 (1500, 0, 0) and the
endemic equilibrium point with largest infective population
𝐸2 (557.678, 861.551, 161.682) are stable and the equilibrium
point 𝐸1 (1170.057, 301.662, 62.589) is unstable.
Figure 2 corresponds to the situation when 𝑅0 > 1 and
also 𝛿 > 1 and in this case we get unique stable endemic
equilibrium point and unstable disease free equilibrium point
𝐸0 . Figure 3 shows the stability of unique feasible disease free
equilibrium 𝐸0 (1333.3, 0, 0) for 𝑅0 < 𝑅0∗ < 1. For 𝛿 < 1, and
𝑅0 ≥ 1 we get two feasible equilibria: the unstable disease
free equilibria and a unique endemic equilibrium which is
shown in Figure 4. This fact is more clear from the bifurcation
Figure 5, which is obtained by considering 𝜆 1 as the critical
parameter. The horizontal axis is labelled with the appropriate
value of the reproduction number 𝑅0 corresponding to this
bifurcation parameter 𝜆 1 . For 0 < 𝑅0 < 1 we get stable
infection free equilibrium point 𝐸0 and at 𝑅0 = 1 we get
forward bifurcation, that is, for 𝑅0 > 1 we get unstable disease
free equilibrium and unique stable endemic equilibrium
point. Figure 6 is showing the backward bifurcation for the
parameter values satisfying 𝛿 > 1. This diagram too is
8
Discrete Dynamics in Nature and Society
Infective population (𝐼)
1400
1200
1000
800
600
400
200
0
0
0.5
1
1.5
Reproduction number (𝑅0 )
2
Stable
Figure 5: Forward bifurcation diagram for 𝛿 < 1 for the parameter
values 𝑏1 = 10, 𝜆 1 = 0.0005, 𝛼 = 0.01, 𝜇1 = 0.01, 𝛾 = 0.01, 𝜆 2 =
0.0003, 𝑏2 = 60, 𝜇2 = 0.5.
Appendix
Let 𝑥 󳨃→ 𝑓(𝑥) ∈ 𝑅𝑛 be a 𝐶1 function for 𝑥 in an open set
𝐷 ⊂ 𝑅𝑛 . Consider the differential equation
1400
Infective population (𝐼)
the disease dies out. We find the condition under which the
system exhibits backward bifurcation at 𝑅0 = 1. The stability
of two endemic equilibria in the case of backward bifurcation
for 𝑅0 < 1 is not discussed analytically but we have shown
these by numerical simulations. It is found that the endemic
equilibrium with the largest infective population is stable,
while the endemic equilibrium with less infective population
is unstable. In the case of a unique endemic equilibrium, the
global stability analysis has been performed. A generalization
of the Poincaré-Bendixson criterion has been used. The sufficient conditions that we found are not completely satisfactory,
as they involve the constant of uniform persistence 𝑐. How
to implement the geometric approach to give better stability
conditions for epidemic models with convex incidence rate
appears to be an open and stimulating challenge.
1200
𝑥󸀠 = 𝑓 (𝑥) .
1000
Denote by 𝑥(𝑡, 𝑥0 ) the solution of (A.1) such that 𝑥(0, 𝑥0 ) =
𝑥0 . We take the following two assumptions.
800
600
(H1) There exits a compact absorbing set 𝐾 ⊂ 𝐷.
400
(H2) Equation (A.1) has a unique equilibrium 𝑥 in 𝐷.
200
0
0
0.2
0.4
0.6
0.8
1
1.2
Reproduction number (𝑅0 )
1.4
1.6
1.8
Stable
Unstable
Figure 6: Backward bifurcation diagram for 𝛿 > 1 for the parameter
values 𝑏1 = 240, 𝜆 1 = 0.00012, 𝛼 = 0.08, 𝜇1 = 0.16, 𝛾 = 0.015, 𝜆 2 =
0.0001, 𝑏2 = 550, 𝜇2 = 0.5.
obtained by considering 𝜆 1 as the critical parameter and
horizontal axis is labelled with the appropriate value of the
reproduction number 𝑅0 corresponding to this bifurcation
parameter 𝜆 1 . It is observed that when the reproduction
number 𝑅0 is between 0 to 0.190526, the infection free
equilibrium alone is stable, for 0.1905261 < 𝑅0 < 1 we have
bistability where either the infection free equilibrium is stable
or the equilibrium 𝐸2 is stable. Again by increasing 𝜆 1 we get
𝑅0 > 1 and we have unique stable endemic equilibrium point
and unstable disease free equilibrium 𝐸0 . The equilibrium 𝐸1
when it exists is always saddle. As backward bifurcation is
quite significant in this figure, so it seems that it can arise for
biologically reasonable parameter values too.
6. Conclusions
In this paper, a dengue disease epidemic model with nonlinear incidence has been analyzed. If 𝑅0 < 1, the diseasefree equilibrium 𝐸0 is locally asymptotically stable, that is,
(A.1)
The equilibrium 𝑥 is said to be globally stable in 𝐷 if it
is locally stable and all trajectories in 𝐷 converge to 𝑥. The
following global-stability problem is formulated in [8].
Theorem A.1. Under the assumptions (H1) and (H2), the
global stability of 𝑥 with respect to 𝐷 is implied by its local
stability.
Assumptions (H1) and (H2) are satisfied if 𝑥 is global
stability in 𝐷. For 𝑛 ≥ 2, a Bendixson criterion is a condition
satisfied by 𝑓 which precludes the existence of nonconstant
periodic solution of (A.1). A Bendixson criterion is said to
be robust under 𝐶1 local perturbations of 𝑓 at 𝑥1 ∈ 𝐷 if,
for sufficiently small 𝜖 ≥ 0 and neighborhood 𝑈 of 𝑥1 , it is
also satisfied by 𝑔 ∈ 𝐶1 (𝐷 → 𝑅𝑛 ) such that the support
supp(𝑓 − 𝑔) ⊂ 𝑈 and |𝑓 − 𝑔|𝐶1 < 𝜖, where
󵄨
󵄨󵄨
𝜕𝑔
󵄨 󵄨󵄨 𝜕𝑓
󵄨
󵄨
󵄨󵄨
󵄨
(𝑥)󵄨󵄨󵄨 : 𝑥 ∈ 𝐷} .
󵄨󵄨𝑓 − 𝑔󵄨󵄨󵄨𝐶1 = sup {󵄨󵄨󵄨𝑓 (𝑥) − 𝑔 (𝑥)󵄨󵄨󵄨 + 󵄨󵄨󵄨 (𝑥)−
󵄨󵄨 𝜕𝑥
󵄨󵄨
𝜕𝑥
(A.2)
Such 𝑔 will be called local 𝜖-perturbations of 𝑓 at 𝑥1 . It is
easy to see that the classic Bendixson’s condition div 𝑓(𝑥) < 0
for 𝑛 = 2 is robust under 𝐶1 local perturbations of at each 𝑥1 ∈
𝑅2 . Bendixson’s condition for higher dimensional system that
are 𝐶1 robust are discussed in [8, 10, 11].
A point 𝑥0 ∈ 𝐷 is wandering for (A.1) if there exists a
neighborhood 𝑈 of 𝑥0 and 𝑇 > 0 such that 𝑈 ∩ 𝑥(𝑡, 𝑈) is
empty for all 𝑡 > 𝑇. Thus, for example, all equilibria and limit
points are nonwandering. The following is a version of the
local 𝐶1 Closing lemma of Pugh (see [12, 13]).
Discrete Dynamics in Nature and Society
9
Lemma A.2. Let 𝑓 ∈ 𝐶1 (𝐷 → 𝑅𝑛 ). Assume that all positive
semitrajectories of (A.1) are bounded. Suppose that 𝑥0 is a
nonwandering point of (A.1) and that 𝑓(𝑥0 ) ≠ 0. Then, for each
neighborhood 𝑈 of 𝑥0 and 𝜖 > 0, there exists a 𝐶1 local-𝜖
perturbation 𝑔 of 𝑓 at 𝑥0 such that
(a) supp(𝑓 − 𝑔) ⊂ 𝑈 and
Let 𝑥 󳨃→ 𝑃(𝑥) ( 𝑛2 ) × ( 𝑛2 ) matrix-valued function that is
𝐶 for 𝑥 ∈ 𝐷. Assume that 𝑃−1 (𝑥) exists and is continuous for
𝑥 ∈ 𝐾, the compact absorbing set. A quantity 𝑞2 is defined as
1
1 𝑡
𝑞2 = lim sup sup ∫ 𝜇 (𝐵 (𝑥 (𝑠, 𝑥0 ))) 𝑑𝑠,
𝑡 → ∞ 𝑥0 ∈𝐾 𝑡 0
where
(b) the perturbed system 𝑥󸀠 = 𝑔(𝑥) has a nonconstant
periodic solution whose trajectory passes through 𝑥0 .
The following general global-stability principle is established in [8].
Theorem A.3. Suppose that assumptions (H1) and (H2) hold.
Assume that (A.1) satisfies a Bendixson criterion that is robust
under 𝐶1 local perturbations of 𝑓 at all nonequilibrium
nonwandering points for (A.1). Then 𝑥 is globally stable in 𝐷
provided it is stable.
The main idea of the proof in [8] for Theorem A.3 is
as follows: suppose that system (A.1) satisfies a Bendixson
criterion. Then it does not have any nonconstant periodic
solutions. Moreover, the robustness of the Bendixson criterion implies that all nearby differential equations have
nonconstant periodic solutions. Thus by Lemma A.2, all
of nonwandering points of (A.1) in 𝐷 must be equilibria.
In particular, each omega limit point in 𝐷 must be an
equilibrium. Therefore 𝜔(𝑥0 ) = {𝑥} for all 𝑥0 ∈ 𝐷 since 𝑥
is the only equilibrium in 𝐷.
A method of deriving a Bendixson criterion in 𝑅𝑛 is
developed in [14]. The idea is to show that second compound
equation
𝑧󸀠 (𝑡) =
𝜕𝑓[2]
(𝑥 (𝑡, 𝑥0 )) 𝑧 (𝑡) ,
𝜕𝑥
(A.3)
with respect to a solution 𝑥(𝑡, 𝑥0 ) ⊂ 𝐷 to (A.1), is uniformly
asymptotically stable, and the exponential decay rate of all
solutions to (A.3) is uniform for 𝑥0 in each compact subset
of 𝐷. Here 𝜕𝑓[2] /𝜕𝑥 is second additive compound matrix of
the Jacobian matrix 𝜕𝑓/𝜕𝑥. Generally speaking, for an 𝑛 × 𝑛
matrix 𝐽 = (𝐽𝑖𝑗 ), 𝐽[2] is a ( 𝑛2 ) × ( 𝑛2 ) matrix and in the special
case 𝑛 = 3, one has
𝐽[2] = (
𝐽11 + 𝐽22
𝑎23
−𝐽13
𝐽32
𝐽11 + 𝐽33
𝐽12 ) .
−𝐽31
𝐽21
𝐽22 + 𝐽33
(A.5)
(A.4)
𝜕𝑓[2] /𝜕𝑥 is a ( 𝑛2 ) × ( 𝑛2 ) matrix, and thus (A.3) is a linear
system of dimension ( 𝑛2 ). If 𝐷 is simply connected, the above
mentioned stability property of (A.3) implies the exponential
decay of the surface area of any compact 2𝑑 surface in 𝐷,
which in turn precludes the existence of any invariant simply
closed rectifiable curve in 𝐷, including periodic orbits. The
required uniform asymptotic stability of system (A.3) can be
proved by constructing a suitable Lyapunov function.
𝐵 = 𝑃𝑓 𝑃−1 + 𝑃
𝜕𝑓[2] −1
𝑃 ,
𝜕𝑥
(A.6)
and the matrix 𝑃𝑓 is obtained by replacing each entry 𝑝𝑖𝑗 of 𝑃
by its derivative in the direction of 𝑓, 𝑝𝑖𝑗𝑓 . The quantity 𝜇(𝐵)
is the Lozinskii ̆ measure of 𝐵 with respect to a vector norm
| ⋅ | in 𝑅𝑁, 𝑁 = ( 𝑛2 ), defined by
𝜇 (𝐵) = lim+
ℎ→0
|𝐼 + ℎ𝐵| − 1
,
ℎ
(A.7)
see [14, p. 41]. It is shown in [8] that, if 𝐷 is simply connected,
the condition 𝑞2 < 0 rules out the presence of any orbit
that gives rise to a simple closed rectifiable curve that is
invariant for (A.1), such as periodic orbits, homoclinic orbits,
and heteroclinic cycles. Moreover, it is robust under 𝐶1 local
perturbations of 𝑓 near any nonequilibrium point that is
nonwandering. In particular, the following global stability
result is proved in Theorem 3.5 of [8].
Theorem A.4. Assume that 𝐷 is simply connected and that
assumptions (H1), (H2) hold. Then the unique equilibrium 𝑥
of (A.1) is globally stable in 𝐷 if 𝑞2 < 0.
Acknowledgments
The authors are very grateful to two referees for their valuable
comments and suggestions which led to an improvement of
their original paper. They also thank the Handling Editor
and all the editorial staff for the help and support provided
to them. This work is supported by the NSF of China (no.
11271314) and NSF of Henan Province (no. 112300410058).
References
[1] Y. Xiao and S. Tang, “Dynamics of infection with nonlinear
incidence in a simple vaccination model,” Nonlinear Analysis.
Real World Applications, vol. 11, no. 5, pp. 4154–4163, 2010.
[2] L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics
and Computation, vol. 217, no. 7, pp. 3531–3541, 2010.
[3] Z. Hu, W. Ma, and S. Ruan, “Analysis of SIR epidemic models
with nonlinear incidence rate and treatment,” Mathematical
Biosciences, vol. 238, no. 1, pp. 12–20, 2012.
[4] L. Li, G.-Q. Sun, and Z. Jin, “Bifurcation and chaos in an
epidemic model with nonlinear incidence rates,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1226–1234, 2010.
[5] M. Ozair, A. A. Lashari, I. H. Jung, and K. O. Okosun, “Stability
analysis and optimal control of a vector-borne disease with
nonlinear incidence,” Discrete Dynamics in Nature and Society,
vol. 2012, Article ID 595487, 21 pages, 2012.
10
[6] C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and
Engineering, vol. 1, no. 2, pp. 361–404, 2004.
[7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied
Mathematical Sciences, Springer, New York, NY, USA, 1983.
[8] M. Y. Li and J. S. Muldowney, “A geometric approach to globalstability problems,” SIAM Journal on Mathematical Analysis, vol.
27, no. 4, pp. 1070–1083, 1996.
[9] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students,
vol. 14 of Software, Environments, and Tools, SIAM, Philadelphia, Pa, USA, 2002.
[10] M. Y. Li and J. S. Muldowney, “On R. A. Smith’s autonomous
convergence theorem,” The Rocky Mountain Journal of Mathematics, vol. 25, no. 1, pp. 365–379, 1995.
[11] Y. Li and J. S. Muldowney, “On Bendixson’s criterion,” Journal
of Differential Equations, vol. 106, no. 1, pp. 27–39, 1993.
[12] C. C. Pugh, “An improved closing lemma and a general density
theorem,” American Journal of Mathematics, vol. 89, pp. 1010–
1021, 1967.
[13] C. C. Pugh and C. Robinson, “The 𝐶1 closing lemma, including
Hamiltonians,” Ergodic Theory and Dynamical Systems, vol. 3,
no. 2, pp. 261–313, 1983.
[14] W. A. Coppel, Stability and Asymptotic Behavior of Differential
Equations, D. C. Heath and Co., Boston, Mass, USA, 1965.
Discrete Dynamics in Nature and Society
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014